Sylvester equation

The Sylvester equation is in mathematics and control theory, a matrix equation of the form

In this case, there are four matrices; are prescribed; is sought.

Can even be a common matrix; then a matrix such as a matrix, and a matrix.

It is named after James Joseph Sylvester, who also published in 1884.

The important applications for the special case in which the adjoint matrix to is also called Lyapunov equation (after Alexander Mikhailovich Lyapunov ).

Existence and uniqueness of the solution

Because of the noncommutativity of the matrix product, the equation can not be resolved directly. Nevertheless, it is simply a linear equation which is a linear system of equations with the unknown, written in vectorized form matrix elements.

In compact form it can be written using the Kronecker product and the Vektorisierungsoperator as follows:

This is the unit matrix.

The direct solution of this equation system is complex ( elements in a sparse matrix, the unknown and flops) and beyond numerically unstable.

The solution exists if and is unique if and have no common eigenvalues.

Numerical resolution

Classically, the solution is calculated stable and robust with the Bartels - Stewart algorithm. Here, and brought by similarity transformations into the Schur normal form, while the Sylvester equation in a simpler and backward substitution transforms detachable triangular shape. The similarity transformations carried out with the derived from the QR algorithm Francis algorithm.

;; and are suitable triangular matrices ( In the real insulated Subdiagonalelemente they may contain ).

Here, and.

In the simpler triangular shape can now be determined directly on and off. The computation time is in the order of the Schur normal form ( FLOPS ).

Newer algorithms come from a Schur transformation (eg B) and, together with the other matrix (eg A) only a Hessenberg matrix.

Even with the iterative solvers for linear systems, the solution can be calculated.

Credentials

  • J. Sylvester: Sur l' s equations matrices. In: C. R. Acad. Sc. Paris. 99, 1884, pp. 67-71, 115-116.
  • RH Bartels, GW Stewart: Solution of the matrix equation. In: Communications of the ACM. 15, No. 9, 1972, p 820-826, doi: 10.1145/361573.361582.
  • R. Bhatia, P. Rosenthal: How and Why to Solve the Operator Equation. In: Bulletin of the London Mathematical Society. 29, No. 1, 1997, pp. 1-21, doi: 10.1112/S0024609396001828.
  • Sang- Gu Lee, Quoc Phong Vu: Simultaneous solutions of Sylvester equations and idempotent matrices Separating the joint spectrum. In: Linear Algebra and its Applications. 435, No. 9, 2011, pp. 2097-2109, doi: 10.1016/j.laa.2010.09.034.
  • Zhou Wang and Qiang Jituan Ruirui Niu: A Preconditioned Iteration Method for Solving Sylvester Equations. 2012.http :/ / www.hindawi.com/journals/jam/2012/401059/
Identity matrix Triangular matrix Digital Object Identifier
758133
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